A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

4, 0, 3, 7, 8, 1, 1, 6, 3, 9, 9, 5, 6, 8, 4, 6, 4, 3, 1, 1, 6, 8, 0, 2, 8, 8, 7, 9, 9, 9, 7, 8, 6, 4, 9, 3, 0, 1, 3, 6, 0, 8, 3, 9, 9, 3, 4, 0, 8, 8, 0, 6, 8, 5, 7, 8, 6, 3, 4, 9, 6, 1, 5, 9, 8, 9, 7, 7, 7, 3, 8, 3, 7, 8, 6, 5, 3, 1, 9, 4, 7, 4, 4, 4, 0, 7, 7, 0, 1, 5, 0, 7, 0, 3, 3, 7, 9, 1, 9, 6, 9, 1, 0, 5, 7

Offset: 1

Author

Amiram Eldar, Oct 01 2024

Keywords

• nonn
• cons

Comments

A point mass is attached to a frictionless pivot by a massless string of length L and revolves in a vertical circle about the pivot in a uniform gravitational field with an acceleration g. The slowest possible motion occurs when the tension in the string is momentarily zero at the top of the route, and the longest-possible period is then c * sqrt(L/g), where c is this constant.

Examples

			4.03781163995684643116802887999786493013608399340880...
		

Links

• Physics StackExchange, Maximum period of a vertically spinning ball, 2016.
• Physics StackExchange, Maximal time for an object vertically swinging on a rope to complete a full circle, 2023.
• Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.
• Wikipedia, Elliptic integral: Complete elliptic integral of the first kind.

Crossrefs

Constants related to similar physical problems: A019692, A038533, A038534, A175574, A256514, A309893, A310000.

Programs

Mathematica

RealDigits[4 * EllipticK[4/5] / Sqrt[5], 10, 120][[1]]

PARI

4*ellK(sqrt(4/5))/sqrt(5)

Formula

Equals 2 * Integral_{0..Pi} (1/sqrt(3 + 2*cos(x))) dx.